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Born 5 March 1915 Paris, France Died 4 July 2002 Paris, France Summary Laurent Schwartz was a French mathematician who is best known for his work in the theory of distributions.

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Laurent Schwartz came from a Jewish background. His father Anselme Schwartz (1872-1957) was born in Balbronn, near Westhoffen, in Alsace shortly after the Franco-Prussian War of 1870-71 which resulted in Alsace being annexed by Germany. He was French và did not like the idea of living in Germany so, at the age of fourteen he left his home town & went lớn Paris where he became a surgeon. He married his first cousin, Claire Debré (1888-1972), the daughter of a rabbi, in 1907. Although Anselme was brought up in the Jewish faith, he became an atheist and brought his children up as atheists. Their family contained many brilliant people such as Claire"s brother, Professor Robert Debré (1882-1978), the founder of Unicef, and Robert Debré"s son Michel Debré (1912-1996), who became a highly successful politician being Prime Minister of France in 1959-62. There was also leading mathematicians in the extended family; Jacques Hadamard was married lớn a sister of Claire Debré"s mother. Laurent was the oldest of his parents three sons, having brothers Daniel (born 1917) và Bertrand (born 1919).When he was eleven years old, Laurent contracted polio. Although he recovered in a few months, the disease left him rather weak for the whole of his life. In September 1926, when he was still in the recovery phase of polio, his parents bought a country house at Autouillet. It was a large house with magnificent gardens surrounded by meadows & fields in which the children could play. When Laurent was young, the family would spend every weekend at Autouillet but lived in Paris during the week. At the lycée he attended in Paris, Schwartz excelled at both mathematics và the classical languages of Greek & Latin. He was faced with a difficult choice, particularly after he was placed first in the national "concours général" in Latin, & fourth in translating. He had to lớn make a choice between spending his final school years studying philosophy và humanities to lớn prepare for university studies in the classics or taking mathematics và philosophy. His mother, who had always played an important role in encouraging her son khổng lồ study, asked her brother Robert Debré for advice. His medical expertise was in the care of children & he gave his professional opinion that Schwartz should study mathematics. This, together with the same opinions from his teachers at the lycée, led Schwartz to lớn decide that he would drop Latin, study both mathematics and philosophy & take the baccalaureate in both subjects. In his final years at the lycée Janson de Sailly he took courses in mathematics & philosophy. He fell in love with geometry, taught by an inspiring teacher, but was disappointed by the philosophy course where the teaching was much less good. Also at the lycée Janson de Sailly he fell in love with Marie-Hélène Levy, who was in the class above him. Marie-Hélène was the daughter of Paul Lévy.Schwartz entered the École Normale Supérieure in Paris in 1934 where he was taught by some of the leading mathematicians in the world. He became engaged to lớn Marie-Hélène, who also studied mathematics, in 1935. Being friendly with her father, Paul Lévy, was an important mathematical influence on the young man. Lévy, whose principle interests were in probability theory & functional analysis, gave the young man a love of these topics which would become his main research interests throughout his life. University is often a time for students khổng lồ become involved in politics and, indeed, Schwartz was active with left-wing beliefs. His political activities at this time are described in <1>:-The intellectual ferment of these years was paralleled by political engagement. Though from a traditionally right-wing background, he was a strong supporter of Leon Blum"s Popular Front Government until he became disillusioned by its failure to support the Spanish Republicans. Similarly, his sympathies for communism were soon dampened by Stalin"s show trials, though he then spent ten years as a Trotskyite, up khổng lồ 1947. He claimed never to regret this, even though it almost prevented him travelling lớn America lớn receive the Fields Medal.He graduated, after an outstanding undergraduate career, with the Agrégation de Mathématiques in 1937. At this stage he decided that he would bởi vì his compulsory military service rather than delay it. He was assigned to the D.C.A. (Défense Contre Aéronefs - Defense Against Aircraft) but, being physically weak and lacking dexterity, he found that he was unable lớn perform tasks such as dismantling & assembling machine guns. He was posted first khổng lồ Ballancourt, then to Biscarrosse, and although he had to lớn extend his military service because of World War II, he never saw combat & was demobbed in August 1940. During this time, in 1938, he had married Marie-Hélène. In fact they had planned to marry in December 1935 but Marie-Hélène contracted a pulmonary tubercular infection & was sent to lớn a sanatorium at Passy, Haut Savoie. The forced separation of eighteen months, during which time they could only correspond by letter, had been extremely difficult for both of them. After Schwartz left military service in 1940 he went with his wife khổng lồ Toulouse where his parents had moved following the German invasion and the fall of France.While in Toulouse, he met Henri Cartan when he visited there to conduct an oral on behalf of the École Normale Supérieure. In fact Marie-Hélène also took the opportunity khổng lồ talk lớn Henri Cartan since she wanted to lớn resume her mathematical studies. Cartan advised him khổng lồ study for a doctorate at Clermont-Ferrand which is where the University of Strasbourg moved after the German armies invaded France at the start of World War II. Schwartz became a member of the Caisse National des Sciences (which later became the CNRS) which supported him until the kết thúc of 1942. After this tư vấn ended, he received funds from the ARS (Aid à la Recherche Scientifique), which supported him to the over of the war. Schwartz received mathematical advice from Georges Valiron who was based in Paris. He had known Valiron since attending his course Functions of a complex variable while he was an undergraduate.Schwartz"s thesis Étude des sommes d"exponentielles Study of sums of exponentials">Ⓣ(Study of sums of exponentials), submitted to lớn the University of Strasbourg in 1943, contains the following acknowledgement as to the help that Valiron had given:-I want khổng lồ especially thank Georges Valiron who not only gave me much advice, but also, through the correspondence he kindly entered into with me, helped me to lớn overcome many difficulties.Valiron was an examiner of the thesis, as was Charles Ehresmann và Andre Roussel. Schwartz also wrote in his thesis:-I would also like to express my gratitude to Jacques Hadamard and Paul Lévy who have guided & enriched my mathematical development.During the war his political activities, Trotskyist beliefs & Jewish background put him in all manner of delicate situations. He adopted a false identity, calling himself Laurent-Marie Sélimartin, & only by a combination of skill và good luck did he escape detection. However, his weak physical condition meant that he was unable to lớn assist the Resistance movement. He had been a staunch supporter of the Trotskyist các buổi tiệc nhỏ since his days as a student at the École Normale Supérieure but his feelings began to change during the war A Mathematician Grappling With His Century (Springer, New York, 2001)." href="../../Biographies/Schwartz/#reference-4">4>:-Trotskyism gave me, during my years at the ENS, a remarkable education, clearly more advanced and sophisticated than that of most youngsters of my age. But by the extremism và sectarianism of its ideas, và by its stereotyped language, it neutralised me during the occupation. My judgment remains extremely severe on my own actions as well as those of the majority of the Trotskyist các buổi tiệc nhỏ during that period.In March 1943 the Schwartz"s son Marc-André was born; this only increased their danger. Marc-André went on to lớn become a writer & poet but had a tragic life - see below. Schwartz spent the academic year 1944-45 lecturing at the Faculty of Science at Grenoble before moving to Nancy where, on the recommendation of Jean Delsarte and Jean Dieudonné, he became a professor in the Faculty of Science. It was during this period of his career that he produced his famous work on the theory of distributions. We described this idea in more details below but, at this point, we quote Schwartz"s own mô tả tìm kiếm of coming up with the idea in 1944 A Mathematician Grappling With His Century (Springer, New York, 2001)." href="../../Biographies/Schwartz/#reference-4">4>:-In my youth I used lớn have insomnias lasting several hours và never took sleeping pills. I remained in my bed, the light off và without writing, did mathematics. My inventive energy was redoubled và I advanced rapidly without tiring. I felt entirely free, without any of the brakes imposed by my daily life & writing. After some hours ... Especially if an unexpected difficulty came up ... I would stop và sleep until morning. I would be tired but happy for the whole of the following day. ... On this particular night I felt sure of myself & filled with a sense of exaltation. I lost no time in rushing to lớn explain everything to lớn Henri Cartan who ... Lived next door. He was enthusiastic: "There you are. You"ve just resolved all the difficulties of differentiation. Now we"ll never again have functions without derivatives".He also continued lớn be politically active, standing in the French elections of 1945 as a Trotskyist and, after failing to be elected, he again stood (also unsuccessfully) in 1946 when in Nancy. During his time in Nancy, he taught a remarkable collection of students, including Bernard Malgrange, Jacques-Louis Lions, François Bruhat, and Alexander Grothendieck. Also during these years he became an international star in the mathematical world, lecturing, at the invitation of Harald Bohr, in Copenhagen in October 1947 và was one of four main speakers at the First Canadian Mathematical Congress in Vancouver in 1949. The Schwartz"s daughter, Claudine, was born in 1947; she became a mathematician and is the tác giả of Gaz. Math. No. 113 (2007), 113-118." href="../../Biographies/Schwartz/#reference-42">42>. (Claudine married the mathematician Raoul Robert in 1971.) In 1953 Schwartz"s wife, Marie-Hélène Schwartz, was awarded a doctorate by the University of Paris for her thesis Formules apparentées à celles de Gauss-Bonnet et Nevanlinna-Ahlfors pour certaines applications d"une variété à n dimensions dans une autre Formulas related to lớn those of Gauss-Bonnet-Nevanlinna và Ahlfors for certain maps of an n-dimensional manifold to another">Ⓣ(Formulas related to lớn those of Gauss-Bonnet-Nevanlinna & Ahlfors for certain maps of an n-dimensional manifold khổng lồ another). Valiron was an examiner of her thesis as he had been for her husband"s ten years earlier.In 1953 Schwartz returned lớn Paris where he became professor, holding this position until 1959. He taught at the École Polytechnique in Paris from 1959 to lớn 1980. He then spent three years at the University of Paris VII before he retired in 1983. We say a little below about his remarkable mathematical contributions but before we look at these we recount some of the political activity he took part in during his career in Paris.In 1956 he was one of the leaders of protests in France against the Russian invasion of Hungary. Then in the following year he became involved in an event much closer to him personally, the "Audin Affair" in Algeria <1>:-Audin, a mathematician & communist based in Algiers, was writing his thesis under Schwartz"s supervision. But in June 1957 the 25-year-old father of three và opponent of French rule in Algeria was abducted by paratroopers, tortured và killed. Schwartz was tireless in his calls for justice, & organised a presentation of the young man"s thesis in his absence.Vocal in his opposition lớn the French campaign, he signed the famous "Declaration des 121" in favour of military insubordination. The riposte of Pierre Messmer, the Minister for the French Army (and, by the same token, of the École), was to strip him of his position at the Polytechnique, for reasons of "common sense và honour". To which Schwartz replied that since the Army commanded by Messmer had sanctioned torture và promoted torturers, such remarks were absurd.After a brief exile in New York, he regained his post two years later ...The outstanding contribution lớn mathematics which Schwartz made in the late 1940s was his work in the theory of distributions. Above, we gave his own mô tả tìm kiếm of the night he came up with the idea. The first publication in which he presented these ideas was Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques Generalization of the notion of function, derivative, Fourier transform và mathematical & physical applications">Ⓣ(Generalization of the notion of function, derivative, Fourier transform và mathematical and physical applications) which appeared in 1948. The theory of distributions is a considerable broadening of the differential & integral calculus. Heaviside và Dirac had generalised the calculus with specific applications in mind. These, and other similar methods of formal calculation, were not, however, built on an abstract và rigorous mathematical foundation. Schwartz"s development of the theory of distributions put methods of this type onto a sound basis, và greatly extended their range of application, providing powerful tools for applications in numerous areas.The lectures he gave in Vancouver in 1949 became the basis for Schwartz"s two-volume treatise Théorie des distributions (1950, 1951). Irving Segal writes in a review:-This is a generally clear, carefully organized, và detailed account of the basic aspects of the theory of "distributions"" due lớn the author, và described by him in earlier publications ... This theory provides a convenient formalism for many common situations in theoretical và applied analysis, but its greatest significance may be in connection with partial differential equations, particularly those of hyperbolic type, where its adaptability to local problems gives it an advantage over Hilbert space (and other primarily global) techniques.In the article on "Analysis" in Encyclopaedia Britannica François Treves describes Schwartz"s work as follows:-... Schwartz"s idea (in 1947) was khổng lồ give a unified interpretation of all the generalized functions that had infiltrated analysis as (continuous) linear functionals on the space Cç of infinitely differentiable functions vanishing outside compact sets. He provided a systematic and rigorous description, entirely based on abstract functional analysis và on duality. It is noteworthy that such an approach had a precedent, in the presentation by André Weil of the integration of locally compact groups ... Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated. This has led lớn extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables. Schwartz"s ideas can be applied khổng lồ many other spaces of test-functions beside Cç, as he himself & others have shown ...Harald Bohr presented a Fields Medal to Schwartz at the International Congress of Mathematicians in Harvard on 30 August 1950 for his work on the theory of distributions. Harald Bohr Proceedings of the International Congress of Mathematicians, Harvard, 1950 (American Mathematical Society, Providence, RI, 1952)." href="../../Biographies/Schwartz/#reference-7">7> described Schwartz"s 1948 paper as one:-... Which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus khổng lồ which the theory leads and on understanding how essential an advance its application may mean khổng lồ many parts of higher analysis, such as spectral theory, potential theory, và indeed the whole theory of linear partial differential equations ...Schwartz has received a long list of prizes, medals & honours in addition to the Fields Medal. He received prizes from the Paris Academy of Sciences in 1955, 1964 and 1972. In 1972 he was elected a member of the Academy. He has been awarded honorary doctorates from many universities including Humboldt (1960), Brussels (1962), Lund (1981), Tel-Aviv (1981), Montreal (1985) and Athens (1993).Later work by Schwartz on stochastic differential calculus is described by him in the survey article Analyse Mathématique et Applications (Gauthier-Villars, Montrouge, 1988), 445-463." href="../../Biographies/Schwartz/#reference-45">45>, see also Mathematical analysis and applications A (Academic Press, New York-London, 1981), 1-25." href="../../Biographies/Schwartz/#reference-44">44>. Later political campaigns include those against American involvement in Vietnam, the Soviet invasion of Afghanistan, & the Russian war against Chechnya. His political activities led, however, khổng lồ a family tragedy Mathematical lives (Springer, Berlin, 2011), 157-164." href="../../Biographies/Schwartz/#reference-21">21>:-His son Marc-André committed suicide in 1971, the result of the life-long trauma following his kidnapping at the hands of French nationalists seeking revenge on his father for his commitment to anti-colonialism and tư vấn of the Algerians seeking independence.With such involvement in mathematics & politics one might imagine that Schwartz would not have had time for a major hobby. This however would be entirely wrong for he was an avid collector of butterflies, with over 20,000 specimens.Among several other books which Schwartz has written, we mention Méthodes mathématiques pour les sciences physiques Mathematical methods for the physical sciences">Ⓣ(Mathematical methods for the physical sciences) (1961). It was reviewed by George Temple who wrote The Mathematical Gazette 46 (357) (1962), 255." href="../../Biographies/Schwartz/#reference-48">48>:-Those who have been privileged lớn see the notes of the lectures which Professor Schwartz delivered in British Columbia a few years ago will know that he is a master of clear, precise exposition which can be readily adapted to lớn the needs of mathematical physicists. The title of this volume has clearly been chosen with some care, khổng lồ indicate that it treats of mathematical methods involved in modern mathematical physics, and that it is not necessarily an introductory text-book for the mathematical physicist. The book does give, in fact, a splendid introduction khổng lồ a number of basic topics in mathematical physics, treated with a degree of rigour and abstraction which may well surprise the physics student at British universities. ... The whole book is written in a concise and lucid style which we have learnt khổng lồ associate with the name of Professor Schwartz.Let us kết thúc by giving two quotes from Schwartz; the first on politics and the second on mathematics:-I have always thought that morality in politics was something essential, just like feelings và affinities.To discover something in mathematics is khổng lồ overcome an inhibition và a tradition.

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